
\(f \sim g \)  if and only if \((f - g)\in \sum _{i > 0}{\rm Im}(Sq^{i})\) . (Note that since working mod 2, \(f-g = f+g.\) )

\(f \sim _{\omega } g\)  if and only if \(f, \, g\in (P_t)_n^{\omega }\)  and
\((f -g)\in (((P_t)_n^{\omega }\cap \sum _{i > 0}{\rm Im}(Sq^{i})) + (P_t)_n^{< \omega }).\)
In particular, if \(f\sim 0\)  (resp. \(f\sim _{\omega } 0\) ), then we say that \(f\)  is hit (resp. \(\omega \) -hit). It is straightforward to see that the above binary relations are equivalence ones. So, one has a quotient space
\((Q_n^{\otimes t})^{\omega } := (P_t)_n^{\omega }/ (( (P_t)_n^{\omega })\cap \sum _{i > 0}{\rm Im}(Sq^{i})) + (P_t)_n^{< \omega }),\) 
from which it is not difficult to check the isomorphism \((QP_t)_n\cong \bigoplus _{\deg (\omega ) = n}(Q_n^{\otimes t})^{\omega }\)  (see [1]}).


